Idea

The subject of topology deals with the expressions of continuity and boundary, and studying the geometric properties of (originally, metric) spaces and relations of subspaces, which do not change under continuous deformations, regardless to the changes in their metric properties. Topology as a structure enables one to model continuity and convergence locally. More recently, in metric spaces, topologists and geometric group theorists started looking at asymptotic properties at large, which are in some sense dual to the standard topological structure and are usually referred to as coarse topology.

There are many cousins of topological spaces, e.g. sites, locales, topoi, higher topoi, uniformity spaces and so on which specialize or generalize some aspect or structure usually found in Top. One of the tools of topology, homotopy theory, has long since crossed the boundaries of topology and applies to many other areas, thanks to many examples and motivations as well as of abstract categorical frameworks for homotopy like Quillen model categories, Brown’s categories of fibrant objects and so on.